On the Balog-Ruzsa Theorem in short intervals

In this paper we give a short interval version of the Balog-Ruzsa theorem concerning bounds for the $L_1$ norm of the exponential sum over $r$-free numbers. As an application, we give a lower bound for the $L_1$ norm of the exponential sum defined with the Möbius function. Namely we show that $\int_{\mathbb T} \left|\sum_{|n-N|<H} μ(n)e(n α)\right| d α\gg H^{\frac{1}{6}}$ when $H \gg N^{\frac{9}{17} + \varepsilon}$.

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